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Question Number 5579 by Rasheed Soomro last updated on 21/May/16

Without using a calculator, evaluate (log 5)^2 +log 2 log 50

$$\mathrm{Without}\:\mathrm{using}\:\mathrm{a}\:\mathrm{calculator},\:\mathrm{evaluate} \\ $$$$\left(\mathrm{log}\:\mathrm{5}\right)^{\mathrm{2}} +\mathrm{log}\:\mathrm{2}\:\mathrm{log}\:\mathrm{50} \\ $$

Answered by FilupSmith last updated on 21/May/16

log x = log_(10) x log(5)^2 +log(5)log(50) =log(5)(log(5)+log(50)) =log(5)log(250) log(250)=log(100×2.5) =log(100)+log(2.5) =2+log(2.5) ∴log(5)log(250)=log(5)(2+log(2.5)) =2log(5)+log(5)log(2.5) continue

$$\mathrm{log}\:{x}\:=\:\mathrm{log}_{\mathrm{10}} {x} \\ $$$$ \\ $$$$\mathrm{log}\left(\mathrm{5}\right)^{\mathrm{2}} +\mathrm{log}\left(\mathrm{5}\right)\mathrm{log}\left(\mathrm{50}\right) \\ $$$$=\mathrm{log}\left(\mathrm{5}\right)\left(\mathrm{log}\left(\mathrm{5}\right)+\mathrm{log}\left(\mathrm{50}\right)\right) \\ $$$$=\mathrm{log}\left(\mathrm{5}\right)\mathrm{log}\left(\mathrm{250}\right) \\ $$$$\mathrm{log}\left(\mathrm{250}\right)=\mathrm{log}\left(\mathrm{100}×\mathrm{2}.\mathrm{5}\right) \\ $$$$=\mathrm{log}\left(\mathrm{100}\right)+\mathrm{log}\left(\mathrm{2}.\mathrm{5}\right) \\ $$$$=\mathrm{2}+\mathrm{log}\left(\mathrm{2}.\mathrm{5}\right) \\ $$$$\therefore\mathrm{log}\left(\mathrm{5}\right)\mathrm{log}\left(\mathrm{250}\right)=\mathrm{log}\left(\mathrm{5}\right)\left(\mathrm{2}+\mathrm{log}\left(\mathrm{2}.\mathrm{5}\right)\right) \\ $$$$=\mathrm{2log}\left(\mathrm{5}\right)+\mathrm{log}\left(\mathrm{5}\right)\mathrm{log}\left(\mathrm{2}.\mathrm{5}\right) \\ $$$${continue} \\ $$

Commented by nchejane last updated on 21/May/16

It is like it is not copied correctly (log5)^2 +log2log50 =(log5)^2 +log2(log(5×10)) =(log5)^2 +log2(log5+log10) =(log5)^2 +log2log5+log2 =(log5)[log5+log2]+log2 =log5log10+log2 =log5+log2 =log10 =1

$${It}\:{is}\:{like}\:{it}\:{is}\:{not}\:{copied}\:{correctly} \\ $$$$\left({log}\mathrm{5}\right)^{\mathrm{2}} +{log}\mathrm{2}{log}\mathrm{50} \\ $$$$=\left({log}\mathrm{5}\right)^{\mathrm{2}} +{log}\mathrm{2}\left({log}\left(\mathrm{5}×\mathrm{10}\right)\right) \\ $$$$=\left({log}\mathrm{5}\right)^{\mathrm{2}} +{log}\mathrm{2}\left({log}\mathrm{5}+{log}\mathrm{10}\right) \\ $$$$=\left({log}\mathrm{5}\right)^{\mathrm{2}} +{log}\mathrm{2}{log}\mathrm{5}+{log}\mathrm{2} \\ $$$$=\left({log}\mathrm{5}\right)\left[{log}\mathrm{5}+{log}\mathrm{2}\right]+{log}\mathrm{2} \\ $$$$={log}\mathrm{5}{log}\mathrm{10}+{log}\mathrm{2} \\ $$$$={log}\mathrm{5}+{log}\mathrm{2} \\ $$$$={log}\mathrm{10} \\ $$$$=\mathrm{1} \\ $$

Commented by Rasheed Soomro last updated on 21/May/16

G^(OO^(v) ) D!

$$\mathbb{G}^{\overset{\mathrm{v}} {\mathcal{OO}}} \mathbb{D}! \\ $$

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